An algorithm producing hyperbolicity equations for a link complement in \(S^ 3\)
An orientable non-compact hyperbolic 3-manifold of finite volume can be decomposed into a finite number of ideal tetrahedra with faces glued together in pairs. These tetrahedra are parametrized by complex numbers satisfying certain rational equations depending on the combinatorics of the gluing. The author produces an algorithm for writing down these hyperbolicity equations for a class of link complements in the 3-sphere. His algorithm refines one described in work of W. Menasco. The class of links to which the algorithm applies are those with a planar projection to \(R^ 2\) which satisfies the following conditions: it is alternating, any two edges with the same endpoints bound a region, its complementary regions are discs and the unbounded region has at least 3 vertices, and it is not the obvious projection of an \((n,2)\) torus knot or link or the unknotted circle. The author shows that by adding additional circles, any link can be enlarged to one to which the algorithm applies. The algorithm is carefully explained with 54 accompanying figures.
orientable non-compact hyperbolic 3-manifold of finite volume
scientific article; zbMATH DE number 147288